Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 0101 1010 - 010 1101 1000 1010 0000 0010 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 0101 1010 - 010 1101 1000 1010 0000 0010: 32 bit single precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)
1. Identify the elements that make up the binary representation of the number:
The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
1
The next 8 bits contain the exponent:
0101 1010
The last 23 bits contain the mantissa:
010 1101 1000 1010 0000 0010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0101 1010(2) =
0 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =
0 + 64 + 0 + 16 + 8 + 0 + 2 + 0 =
64 + 16 + 8 + 2 =
90(10)
3. Adjust the exponent.
Subtract the excess bits: 2(8 - 1) - 1 = 127,
that is due to the 8 bit excess/bias notation.
The exponent, adjusted = 90 - 127 = -37
4. Convert the mantissa from binary (from base 2) to decimal (in base 10).
The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).
010 1101 1000 1010 0000 0010(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0 + 0.25 + 0 + 0.062 5 + 0.031 25 + 0 + 0.007 812 5 + 0.003 906 25 + 0 + 0 + 0 + 0.000 244 140 625 + 0 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 238 418 579 101 562 5 + 0 =
0.25 + 0.062 5 + 0.031 25 + 0.007 812 5 + 0.003 906 25 + 0.000 244 140 625 + 0.000 061 035 156 25 + 0.000 000 238 418 579 101 562 5 =
0.355 774 164 199 829 101 562 5(10)
5. Put all the numbers into expression to calculate the single precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)1 × (1 + 0.355 774 164 199 829 101 562 5) × 2-37 =
-1.355 774 164 199 829 101 562 5 × 2-37 =
-0.000 000 000 009 864 555 353 122 916 869 779 146 5
1 - 0101 1010 - 010 1101 1000 1010 0000 0010 converted from a 32 bit single precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (float) = -0.000 000 000 009 864 555 353 122 916 869 779 146 5(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
More operations with 32 bit single precision IEEE 754 binary floating point standard representation numbers: